Presenting Functors on Many-Sorted Varieties and Applications
This paper studies several applications of the notion of a presentation of a functor by operations and equations. We show that the technically straightforward generalisation of this notion from the one-sorted to the many-sorted case has several interesting consequences. First, it can be applied to give equational logic for the binding algebras modelling abstract syntax. Second, it provides a categorical approach to algebraic semantics of first-order logic. Third, this notion links the uniform treatment of logics for coalgebras of an arbitrary type T with concrete syntax and proof systems. Analysing the many-sorted case is essential for modular completeness proofs of coalgebraic logics.
A. Kurz and D. Petrişan, “Presenting functors on many-sorted varieties and applications,” Information and Computation, vol. 208, no. 12, pp. 1421–1446, Dec. 2010. DOI: 10.1016/j.ic.2009.11.007